Communication channel optimization systems and methods in multi-user communication systems

ABSTRACT

Systems and methods of optimizing communication channels in multi-user communication systems are provided. Coding weights are determined based on communication channel state information for communication channels between a transmitter and multiple receivers. The coding weights are applied to communication signals to be transmitted from the transmitter to the receivers. Each receiver decodes received signals using inverses of the coding weights. Embodiments of the invention support multi-user MIMO (Multiple Input Multiple Output) where each receiver has fewer antennas than the transmitter, and enhance system performance if the total number of antennas at all of the receivers exceeds the number of antennas at the transmitter.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority from U.S. provisional patentapplication Ser. No. 60/517,389, filed on Nov. 6, 2003, and provisionalpatent application Ser. No. 60/517,893, filed on Nov. 7, 2003. Theentire contents of each of these provisional applications are herebyincorporated herein by reference.

FIELD OF THE INVENTION

The invention relates to communication systems in general, andparticularly to communication channel optimization in multi-usercommunication systems.

BACKGROUND

In any communication system, the quality and capacity of a communicationchannel are affected by such factors as interference, allocation ofcommunication resources, the communication schemes or algorithms used onthe communication channel, and the particular communication equipmentimplemented at transmitting and receiving ends of the channel.

The effects of certain factors may be mitigated through efficientresource allocation and selection of communication schemes andequipment. According to some conventional communication techniques,processing operations intended to compensate for other communicationchannel effects are primarily receiver based. For example, interferencecancellation is performed by a receiver in known communication systems.In addition, the implementation of different types of communicationequipment in conjunction with the same type of channel, such asdifferent communication terminals in a wireless communication system forinstance, may affect received signal processing operations at allreceivers.

Communication terminals at the ends of a communication channel areseldom identical. In wireless communication systems, for example, usercommunication terminals at one end of a communication channel normallyhave much more limited resources than base stations. In known MIMO(Multiple Input Multiple Output) systems, each receiver has at least asmany antennas as a transmitter. This constraint is difficult to satisfywhere communication equipment on opposite ends of a communicationchannel are significantly different, as in the case of wirelesscommunication terminals and base stations in wireless communicationsystems, for example. In addition, resource limitations at one receiverin such a multi-user system can also affect other receivers in thesystem.

SUMMARY OF THE INVENTION

According to one aspect of the invention, a method of processing signalsto be transmitted to receivers on communication channels is provided.The method includes determining pre-coding signal weights based onchannel state information associated with the communication channels toprovide proportional power allocation to the signals, and applying thesignal weights to the signals. In a preferred embodiment, the channelstate information is received from the receivers.

The invention also provides, in another aspect, a method which includesreceiving over a sub-group of communication channels a subset of signalsto which pre-coding signal weights based on channel state informationassociated with the communication channels to provide proportional powerallocation have been applied. The received subset of signals usinginverses of the pre-coding signal weights based on channel stateinformation associated with the sub-group of channels to decode thereceived subset of signals.

According to an embodiment of the invention, the signals to betransmitted include respective groups of signals to be transmitted tothe receivers, and the pre-coding weights are determined to separate therespective groups of signals. Decoding then separates individual signalsin the received subset of signals.

In another aspect, the invention provides a system for processingsignals to be transmitted to receivers on communication channels. Thesystem preferably includes an input for receiving the signals and aprocessor. The processor is configured to determine pre-coding signalweights based on channel state information associated with thecommunication channels to provide proportional power allocation to thesignals, and to apply the signal weights to the signals. In a preferredembodiment, the system further includes multiple antennas which providerespective sub-MIMO channels to the receivers.

The invention also provides a system that includes an input an input forreceiving over a sub-group of communication channels a subset of signalsto which pre-coding signal weights based on channel state informationassociated with the communication channels to provide proportional powerallocation have been applied, and a processor. The processor isconfigured to decode the received subset of signals using inverses ofthe pre-coding signal weights based on channel state informationassociated with the sub-group of the channels.

A further method of processing signals to be concurrently transmitted toreceivers over communication channels, in accordance with still anotheraspect of the invention, includes determining channel state informationfor the communication channels, determining a spatial coding matrixwhich includes respective set of spatial coding weights for each of thereceivers based on the channel state information, and applying thespatial coding weights in the spatial coding matrix to the signals.

A method in accordance with a still further aspect of the inventionincludes determining channel state information for a communicationchannel between a receiver and a transmitter, transmitting the channelstate information to the transmitter, and receiving from the transmitterone of a plurality of demodulation matrices for demodulatingsubsequently received communication signals to which spatial codingweights comprising respective sets of spatial coding weights formultiple receivers have been applied.

A network element for processing signals to be concurrently transmittedto multiple communication terminals in a communication network is alsoprovided. The network element preferably includes an input configured toreceive the signals, and a processor. The processor is configured todetermine channel state information for each communication channelbetween the network element and the communication terminals, todetermine a spatial coding matrix comprising a respective set of spatialcoding weights for each of the communication terminals based on thechannel state information, and to apply the spatial coding weights inthe spatial coding matrix to the signals.

In a related aspect, a communication terminal for operation in acommunication network is provided. A processor in the terminal isconfigured to determine channel state information for communicationchannels between the communication terminal and a network element in thecommunication network. The terminal also includes at least one antennafor transmitting the channel state information from the communicationterminal to the network element, receiving a demodulation matrix fromthe network element, and receiving signals concurrently transmitted tomultiple communication terminals from the network element. The processoris further configured to demodulate the received signals using thedemodulation matrix.

Other aspects and features of the present invention will become apparentto those ordinarily skilled in the art upon review of the followingdescription of the specific embodiments of the invention.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention will now be described in greater detail with reference tothe accompanying diagrams, in which:

FIG. 1 is a block diagram of a 6×6 MIMO system;

FIG. 2 is a block diagram of a decomposed 6×6 MIMO system;

FIG. 3 is a block diagram of a MIMO system, illustrating inter-userinterference;

FIG. 4 is a block diagram of a MIMO system using polarized communicationchannels, illustrating inter-user interference;

FIG. 5 shows a block diagram of a MIMO system according to an embodimentof the invention;

FIG. 6 is a block diagram of a MIMO BLAST system;

FIG. 7 is a block diagram of a multi-user MIMO system in accordance withanother embodiment of the invention;

FIG. 8 is a block diagram of a conventional null beamforming MIMOsystem;

FIG. 9 is a block diagram of a known multi-user BLAST MIMO system;

FIG. 10 is a block diagram of a known round-robin TDM (Time DivisionMultiplexing) BLAST MIMO system;

FIG. 11 is a plot of BLER (Block Error Rate) versus SNR (Signal-to-NoiseRatio) for an embodiment of the invention and several knowncommunication schemes;

FIG. 12 is a plot of BLER versus Eb/No (Energy per Bit to Spectral NoiseDensity ratio) for an embodiment of the invention and several knowncommunication schemes; and

FIG. 13 is a plot of BLER versus SNR for several embodiments of theinvention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

According to embodiments of the present invention, systems and methodsare provided which enhance the performance of communication channels ina communication system, to thereby improve, for example, thetransmission performance of multi-user MIMO (Multiple Input MultipleOutput) communication systems.

In MIMO systems, a multi-data stream transmitter at a base transceiverstation (BTS) that provides communication services for a coverage areaor cell in a wireless communication system transmits communicationsignals to user terminals via multiple antennas. User terminals are alsocommonly referred to as user equipment (UE), communication devices, andmobile stations, for instance. At a UE side, multiple receive antennasare employed for each user.

FIG. 1 is a block diagram of a 6×6 MIMO system, which includes aplurality of antennas 16 at a BTS 10, and a plurality of antennas 18 anda MIMO decoder 20 at a UE 12. In this system, 6 communication signals 14intended for the UE 20, labelled as s₁ ⁽¹⁾ through s₆ ⁽¹⁾ aretransmitted via the antennas 16 from the BTS 10 to the UE 12. At the UE12, each of the antennas 18 receives the signals transmitted from theantennas 16, and the received signals are decoded in a MIMO decoder 20.

It should be appreciated that the system of FIG. 1 is intended forillustrative purposes only. As will be apparent to those skilled in theart to which the present invention pertains, BTSs include furthercomponents in addition to the antennas 16, such as components togenerate the signals s₁ ⁽¹⁾ through S₆ ⁽¹⁾ for instance. Similarly, theUE 12 includes components to further process received signals decoded bythe MIMO decoder 20. Also, the BTS 10 and the UE 12 normally supportboth transmit and receive operations.

Known MIMO systems do not support simultaneous multi-user MIMOtransmissions where each UE does not have at least the same number ofantennas as a BTS. Instead, MIMO is typically used as a single“fat-pipe”, and multiple users are served through the use of timedivision techniques. In addition, it is practically difficult to realizevery large dimension MIMO systems, 8×8 systems for example, due thephysical size limitations of UEs. As the physical size of a UE isusually limited, the distance by which multiple antennas for largerdimension MIMO systems can be separated is also limited, such that theantennas at the UE become highly correlated. This drastically reducesthe MIMO channel capacity.

However, large dimension MIMO systems may be decomposed into combinedsub-MIMO systems, because in general, channel fading between differentUEs is un-correlated. MIMO channel capacity can then be more efficientlyexploited. FIG. 2 is a block diagram of a decomposed 6×6 MIMO system.The decomposed 6×6 MIMO system includes antennas 32 at a BTS 22, andthree UEs 24, 26, 28, each of which includes a pair of antennas 34/36,40/42, 46/48 and a MIMO decoder 38, 44, 50. At the BTS 22, signals s₁⁽¹⁾/s₂ ⁽¹⁾, s₁ ⁽²⁾/s₂ ⁽²⁾, s₁ ⁽³⁾/s₂ ⁽³⁾ are transmitted via arespective antenna in each of a plurality of sub-groups, pairs in FIG.2, of the antennas 32 to corresponding UEs 24, 26, 28. The superscripts(1), (2), (3) indicate for which UE 24, 26, 28, a signal is intended.Each pair of the antennas 32 forms a 2×2 sub-MIMO channel with the pairof antennas of a respective one of the UEs 24, 26, 28.

To apply the MIMO technique to a multi-user system, inter-user MIMOinterference is a major issue. In FIG. 2, for example, although eachpair of signals s₁ ⁽¹⁾/s₂ ⁽¹⁾, s₁ ⁽²⁾/s₂ ⁽²⁾, s₁ ⁽³⁾/s₂ ⁽³⁾ may beintended for a particular UE 24, 26, 28, the antennas 34/36, 40/42,46/48 in each of the UEs 24, 26, 28 receive communication signals fromall of the antennas 32. In order to cancel such interference using knowntechniques, the number of receive antennas must be equal to or greaterthan the number of transmit antennas. Due primarily to physical spaceand form factor constraints on UEs, the number of antennas that can beprovided at a UE is limited. Thus, for downlink (BTS to UE)transmissions, the number of receive antennas is typically smaller thanthe number of transmit antennas.

FIGS. 3 and 4 are block diagrams of MIMO systems, and illustrateinter-user interference. In the system of FIG. 3, the signals 58,labelled s₁ ⁽¹⁾/s₂ ⁽¹⁾, s₁ ⁽²⁾/s₂ ⁽²⁾, are transmitted from the BTS 52via respective ones of each pair of the antennas 60 to the UEs 54, 56.Signals received by the antennas 62/64, 68/70 in the UEs 54, 56 areprocessed by the MIMO decoders 66, 72. Interference in the MIMO systemof FIG. 3 is indicated at 73. As shown, any communication signals thatare received at one of the UEs 54, 56 but intended for the other of theUEs 54, 56 represent interference at that UE. For example, versions ofs₁ ⁽²⁾ and s₂ ⁽²⁾ received at the UE 54 represent interference.

Similarly, in FIG. 4, the signals 80, labelled s₁ ⁽¹⁾/s₂ ⁽¹⁾, s₁ ⁽²⁾/s₂⁽²⁾, are transmitted from the BTS 74 via respective ones of each pair ofthe antennas 82 to the UEs 76, 78, received by the antennas 84/86,90/92, and processed by the MIMO decoders 88, 94. Polarized channels,provided by the vertical- and horizontal-polarization antennas V and Hin the example system of FIG. 4, reduce interference 95 relative to thegeneral MIMO system of FIG. 3, but do not cancel the inter-userinterference to such a degree that interference cancellation isunnecessary.

The task of interference cancellation is typically performed at a userterminal. In accordance with an embodiment of the invention, downlinkcommunication channel interference cancellation is effectively splitbetween the BTS (transmit) side and the UE (receive) side. For example,a BTS may perform inter-user separation based pre-coding of data to betransmitted, while at the UE side, a UE performs MIMO layer separationand decoding.

One type of multi-user MIMO system in which the invention may beimplemented delivers communication signals according to the layeredspace-time known as MIMO-BLAST concurrently to multiple users. Such asystem is preferably realized with feedback of communication channelstate information from each UE to a BTS. Channel state feedbacktechniques are very well suited for application in conjunction withfixed or nomadic wireless communication channels due to the slowvariation of such channels, which allows accurate channel stateinformation feedback from the UEs to the BTS.

In one embodiment, a closed-loop pre-coded transmit antenna array forsub-MIMO transmission, preferably MIMO-BLAST transmission, in amulti-user environment is provided. Channel state information ismeasured by or fed back to a BTS, and at the BTS side, jointly optimizedweights are computed and applied to antenna input signals to cancelinter-user MIMO interference. Therefore, in one sense, a system inaccordance with this embodiment of the invention may be considered as anadaptive weighted transmit antenna array operating in the MIMO-BLASTmode. This concept is a departure from the conventional beamformingphased antenna array.

A MIMO system can be expressed asy=H s+ η,   (1)where

-   -   y=[y₁ y₂ . . . y_(N)]^(T) is a vector of communication signals        received at a receiver;    -   s=[s₁ s₂ . . . s_(M)]^(T) is a vector of communication signals        transmitted by a transmitter;    -   η=[η₁ η₂ . . . η_(N)]^(T) is a vector of noise components        affecting the transmitted communication signals;

$H = \begin{bmatrix}h_{11} & h_{12} & \ldots & h_{1M} \\h_{21} & h_{22} & \ldots & h_{2M} \\\vdots & \vdots & ⋰ & \vdots \\h_{N1} & h_{N2} & \ldots & h_{NM}\end{bmatrix}$is a channel matrix of communication channel attenuation factors;

-   -   N is a number of antennas at the receiver; and    -   M is a number of antennas at the transmitter.

To decode the transmitted signal s, the receiver performs the inverseprocesss=G y−G η,   (2)whereG=H ⁺=(H′H)⁻¹ H′  (3)is the Moore-Penrose pseudo-inverse of H, and H′ is a conjugate matrixof H, illustratively a Hermitian conjugation or conjugate transpose. Thepost-detection SNR (signal-to-noise ratio) for a decoded element s_(i)of s is given by

$\begin{matrix}{{\gamma_{i} = \frac{{s_{i}}^{2}}{\sigma^{2}{{\overset{\rightarrow}{g}}_{i}}^{2}}},} & (4)\end{matrix}$where g _(i) is the ith row of G, and σ² represents the variance of theelements in η (assuming that all the elements in η have the same levelof variance). From equation (4), it can be seen that the SNR ofpost-detection signals is determined by G, which is given by equation(3). Note that ∥g_(i)∥² is not only determined by channel attenuationfactors |h_(ij)|, but by the condition of the channel matrix H as well.When H is ill-conditioned, ∥g_(i)∥² can be very large; hence γ_(i) willbe very small.

From equation (2), it can be seen that the post-detection signal poweris a fixed value |s_(i)|², such that γ_(i) is in fact determined by thesecond term G η only. System performance can be improved by reducingpost-detection noise, which is represented by ∥g_(i)∥². One possibleissue to consider in this regard is whether, when a transmitter hasinformation about H, pre-equalization can improve system performance.

By defining a new signal vector, illustratively for the square channelmatrix case in which M=N,x=J s=H′(HH′)⁻¹ s,   (5)we haver=H x= s.   (6)

The matrices G and J are equal in this example, when H is a squarematrix (M=N).

The SNR for this pre-equalized signal can be expressed as

$\begin{matrix}{{\kappa_{i} = \frac{{s_{i}}^{2}}{{{\overset{\rightarrow}{j}}_{i}}^{2}\sigma^{2}}},} & (7)\end{matrix}$where j _(i) is the ith row of J.

The pre-equalization approach is similar to power control, i.e., theweak user gets more power so that all the users are equal. However, thisis not an efficient approach to utilize system power.

Another important observation is that the pre-equalization matrix J iscompletely determined by the channel matrix H. That is, except formaking up an identity matrix, there are no other kinds of optimizationsin J.

An embodiment of the invention provides for user separation at thetransmitter. One preferred user separation technique allows the use ofML (Maximum Likelihood) detection schemes at a receiver, such that thediversity order for each receiver is increased. In addition, sincelayers need not be separated within a BTS, system performance may beimproved in at least two further aspects, namely, to enhance anequivalent channel matrix and proportional transmitting powerallocation.

FIG. 5 shows a block diagram of a MIMO system according to an embodimentof the invention. The system includes a BTS 100 having a pre-coder 106and a plurality of antennas 108, 110, 112, 114, and UEs 102, 104, eachhaving a plurality of antennas 116/118, 122/124 and a MIMO decoder 120,126. In FIG. 5, M=4, N_(i)=2 is the number of antennas at an ith UE, U=2is the number of UEs, and M=U*N_(i). In a particularly preferredembodiment, the system of FIG. 5 is a MIMO BLAST system.

Of course, the system of FIG. 5 is one illustrative example of a systemin which the invention may be implemented. The invention is in no waylimited thereto. Extension of the principles of the present invention tosystems having other dimensions will be apparent to those skilled in theart.

At the BTS side, the BTS 100 preferably uses the U degrees of freedom(one per UE 102, 104) of the transmit antennas 108, 110, 112, 114 toperform weighted pre-coding of the signals s₁ ⁽¹⁾, s₂ ⁽¹⁾, s₁ ⁽²⁾, s₂⁽²⁾ in the pre-coder 106, while reserving the N_(i) degrees of freedom(one per receive antenna of each UE 102, 104) of the transmit antennas108, 110, 112, 114 to maximize the N_(i)×N_(i) sub-MIMO channel capacityand proportional antenna power allocation for each user.

At the UE side, each UE 102, 104 decodes communication signals receivedat its antennas 116/118, 122/124, using ML or MMSE (Minimum Mean SquaredError) decoding, for example. The N_(i) antennas also provide diversitygain.

The UE-1 102 preferably determines and feeds back channel stateinformation H₁ (4×2) to the BTS 100. The UE-2 104 similarly preferablydetermines and feeds back channel state information H₂ (4×2) to the BTS100. For some types of communication channel, channel state informationmay instead be determined locally by the BTS 100. Thus, channel stateinformation determination is shown conceptually in FIG. 5 outside theUEs 102, 104. Those skilled in the art will appreciate that channelstate determination may be performed in the UEs 102, 104 or the BTS 100by a digital signal processor (DSP) or a general-purpose processoradapted to execute signal processing software, for example. Varioustechniques for determining channel state information will be apparent tothose skilled in the art, including estimation based on pilot channelsin CDMA (Code Division Multiple Access) systems or preambles andscattered pilot tones in OFDM (Orthogonal Frequency DivisionMultiplexing) systems, for example.

Based on the channel state information H₁ and H₂, the BTS 100 computesan antenna weight matrix or pre-coding matrix P, which preferablycancels inter-user MIMO interference between UE-1 102 and UE-2 104 andmaximizes MIMO system channel capacity for UE-1 102 and UE-2 104.

Before proceeding with a detailed analysis of the system of FIG. 5, acombined (with respect to users) 2×2 MIMO system, in which M=2, N_(i)=1,and U=2, is first considered. In this system, there are no multiplelayers associated with each user. The channel matrix H can be expressedas

$\begin{matrix}{H = {\begin{bmatrix}h_{11} & h_{12} \\h_{21} & h_{22}\end{bmatrix}.}} & (8)\end{matrix}$Now we define a pre-coding matrix P, so that

$\begin{matrix}{C = {{HP} = {{\begin{bmatrix}h_{11} & h_{12} \\h_{21} & h_{22}\end{bmatrix}\begin{bmatrix}p_{11} & p_{12} \\p_{21} & p_{22}\end{bmatrix}}\mspace{25mu} = {\begin{bmatrix}{{h_{11}p_{11}} + {h_{12}p_{21}}} & {{h_{11}p_{12}} + {h_{12}p_{22}}} \\{{h_{21}p_{11}} + {h_{22}p_{21}}} & {{h_{21}p_{12}} + {h_{22}p_{22}}}\end{bmatrix}.}}}} & (9)\end{matrix}$One goal is to identify if a solution to the following equation exists:

$\begin{matrix}\left\{ {\begin{matrix}{\max\limits_{p_{11},p_{21}}{{{h_{11}p_{11}} + {h_{12}p_{21}}}}} \\{{{h_{21}p_{11}} + {h_{22}p_{21}}} = 0}\end{matrix}\left\{ {\begin{matrix}{\max\limits_{p_{12},p_{22}}{{{h_{21}p_{12}} + {h_{22}p_{22}}}}} \\{{{h_{11}p_{12}} + {h_{12}p_{22}}} = 0}\end{matrix}.} \right.} \right. & (10)\end{matrix}$

In the pre-equalization approach, the set of elements in apre-equalization matrix {j₁₁,j₂₁,j₁₂,j₂₂} are used to satisfy thefollowing condition:

$\begin{matrix}{\begin{matrix}\left\{ \begin{matrix}{{{h_{11}j_{11}} + {h_{12}j_{21}}} = 1} \\{{{h_{21}j_{11}} + {h_{22}j_{21}}} = 0}\end{matrix} \right. \\\left\{ \begin{matrix}{{{h_{21}j_{12}} + {h_{22}j_{22}}} = 1} \\{{{h_{11}j_{12}} + {h_{12}j_{22}}} = 0}\end{matrix} \right.\end{matrix}.} & (11)\end{matrix}$Given the same ∥ p _(i)∥²=∥ j _(i)∥², with i=1,2, equation (11) forcesh₁₁j₁₁+h₁₂j₂₁ and h₂₁j₁₂+h₂₂j₂₂ to 1, while equation (10) tries tomaximize the power of the analogous components h₁₁p₁₁+h₁₂p₂₁ andh₂₁p₁₂+h₂₂p₂₂. This illustrates one primary difference between the userseparation techniques according to embodiments of the invention andpre-equalization approaches.

Equation (10) can be manipulated to the form of

$\begin{matrix}{\begin{matrix}\left\{ \begin{matrix}{\max\limits_{p_{11}}{{p_{11}\left( {h_{11} - \frac{h_{12}h_{21}}{h_{22}}} \right)}}} \\{p_{21} = {- \frac{h_{21}h_{11}}{h_{22}}}}\end{matrix} \right. \\\left\{ \begin{matrix}{\max\limits_{p_{22}}{{p_{22}\left( {h_{22} - \frac{h_{21}h_{12}}{h_{11}}} \right)}}} \\{p_{12} = {- \frac{h_{12}h_{22}}{h_{11}}}}\end{matrix} \right.\end{matrix}.} & (12)\end{matrix}$

Although there is no optimized solution to equation (12), selection ofp₁₁ and p₂₂ such that

$\begin{matrix}{{p_{11}} \propto {{{h_{11} - \frac{h_{12}h_{21}}{h_{22}}}}\mspace{14mu}{and}\mspace{14mu}{p_{22}}} \propto {{h_{22} - \frac{h_{12}h_{21}}{h_{11}}}}} & (13)\end{matrix}$improves the system capacity.

If each UE has only one receive antenna and receives only one layer ofthe MIMO signal, then proportional power allocation may be achieved.However, if a UE has multiple antennas and receives multiple layers ofsignals, as in FIG. 5, then the situation will be different.

As described briefly above, the pre-coder 106 at the BTS 100 determinesand applies pre-coding weights to the signals s₁ ⁽¹⁾, s₂ ⁽¹⁾, s₁ ⁽²⁾, s₂⁽²⁾ to perform the function of inter-user interference cancellationthrough user separation. Each receiver, UEs 102, 104 in FIG. 5, thenperforms the function of inter-antenna interference cancellation. Thus,algorithms such as ML and iterative ZF (Zero Forcing)/MMSE can be usedto further improve detection results. The concept of splittinginterference cancellation functions between the transmitter andreceivers is described in further detail below.

For the combined 2×2 MIMO multi-user system of FIG. 5, the pre-codingmatrix P is selected so that the combined channel matrix C has theformat of

$\begin{matrix}{C = {{HP} = {\begin{bmatrix}c_{11} & c_{12} & 0 & 0 \\c_{21} & c_{22} & 0 & 0 \\0 & 0 & c_{33} & c_{34} \\0 & 0 & c_{43} & c_{44}\end{bmatrix}.}}} & (14)\end{matrix}$Note that only c ₁ and c ₂, where c _(i) represents the i-th row of C,affect the UE-1 102, and that only c ₃ and c ₄ affect the UE-2 104. Thecombined channel matrix C has a form of U N_(i)×N_(i) sub-matrices,diagonal elements of which are respective diagonal elements of C.Elements of C outside the plurality of N_(i)×N_(i) sub-matrices arezero. In another sense, C may be considered as having groups of rowsassociated with UEs and groups of columns respectively associated withantenna pairs. For example, c ₁ and c ₂ are associated with the UE-1 102as described above, and the columns of C are respectively associatedwith the antennas 108, 110, 112, 114. As the antenna pairs 108/110 and112/114 provide sub-MIMO channels to the UEs 102 and 104, each pair maybe considered to be associated with a respective one of the UEs 102 and104. Thus, each element of C positioned in a row associated with aparticular UE and a column corresponding to an antenna that isassociated with a different UE is forced to zero by selection ofpre-coding weights in the pre-coding matrix P.

The first two columns of the pre-coding matrix P are determined suchthat they satisfy

$\begin{matrix}\left\{ {\begin{matrix}{c_{11} = {{h_{11}p_{11}} + {h_{12}p_{21}} + {h_{13}p_{31}} + {h_{14}p_{41}}}} \\{c_{21} = {{h_{21}p_{11}} + {h_{22}p_{21}} + {h_{23}p_{31}} + {h_{24}p_{41}}}} \\{{{h_{31}p_{11}} + {h_{32}p_{21}} + {h_{33}p_{31}} + {h_{34}p_{41}}} = 0} \\{{{h_{41}p_{11}} + {h_{42}p_{21}} + {h_{43}p_{31}} + {h_{44}p_{41}}} = 0}\end{matrix},{and}} \right. & (15) \\\left\{ {\begin{matrix}{c_{12} = {{h_{11}p_{12}} + {h_{12}p_{22}} + {h_{13}p_{32}} + {h_{14}p_{42}}}} \\{c_{22} = {{h_{21}p_{12}} + {h_{22}p_{22}} + {h_{23}p_{32}} + {h_{24}p_{42}}}} \\{{{h_{31}p_{12}} + {h_{32}p_{22}} + {h_{33}p_{32}} + {h_{34}p_{42}}} = 0} \\{{{h_{41}p_{12}} + {h_{42}p_{22}} + {h_{43}p_{32}} + {h_{44}p_{42}}} = 0}\end{matrix}.} \right. & (16)\end{matrix}$The second two columns of P, related to the UE-2 104, are preferablydetermined in an analogous manner.

From equation (14), it is not difficult to see that the equivalentsystem for UE-1 102 is

$\begin{matrix}{{\begin{bmatrix}y_{1} \\y_{2}\end{bmatrix} = {\begin{bmatrix}c_{11} & c_{12} \\c_{21} & c_{22}\end{bmatrix}\begin{bmatrix}s_{1} \\s_{2}\end{bmatrix}}},} & (17)\end{matrix}$which represents a 2×2 MIMO system. When an ML decoder is used at a UE,the diversity order is two.

The decoders 120, 126 decode received signals using an inverse matrixsuch as the Moore-Penrose pseudo-inverse matrix of a correspondingsub-matrix of P. In FIG. 5, Q₁ and Q₂ indicate sub-matrices of such aninverse matrix Q that relate to the UEs 102, 104, respectively.Preferably, an inverse matrix D of C=HP, or strictly sub-matrices D₁ andD₂ thereof, are used by the decoders 120, 126 in FIG. 5. These matricescan be derived by each receiver based on channel estimation using pilotchannels, for example. In one embodiment, the BTS 100 sends pilot tonesto UEs 102, 104, and each UE the estimates the elements of itscorresponding inverse sub-matrix.

For brevity, the following analysis relates only to the UE-1 102. Thoseskilled in the art will appreciate that the analysis for the UE-2 104would be similar.

In equation (15), let p₃₁ and p₄₁ force c₁₃=c₁₄=0, which gives

$\begin{matrix}{{\begin{bmatrix}p_{31} \\p_{41}\end{bmatrix} = {- {{{\frac{1}{\Delta}\begin{bmatrix}h_{44} & {- h_{34}} \\{- h_{43}} & h_{33}\end{bmatrix}}\begin{bmatrix}h_{31} & h_{32} \\h_{41} & h_{42}\end{bmatrix}}\begin{bmatrix}p_{11} \\p_{21}\end{bmatrix}}}},} & (18)\end{matrix}$where Δ=h₃₃h₄₄−h₃₄h₄₃.

If A is defined as

$\begin{matrix}{{A = {\begin{bmatrix}h_{11} & h_{12} \\h_{21} & h_{22}\end{bmatrix} - {{{\frac{1}{\Delta}\begin{bmatrix}h_{13} & h_{14} \\h_{23} & h_{24}\end{bmatrix}}\begin{bmatrix}h_{44} & {- h_{34}} \\{- h_{43}} & h_{33}\end{bmatrix}}\begin{bmatrix}h_{31} & h_{32} \\h_{41} & h_{42}\end{bmatrix}}}},{then}} & (19) \\{\begin{bmatrix}c_{11} \\c_{21}\end{bmatrix} = {{A\begin{bmatrix}p_{11} \\p_{21}\end{bmatrix}}.}} & (20)\end{matrix}$As the matrix A is determined by the channel matrix H, p₁₁ and p₁₂ arefree to be chosen, and as such can be used for channel matrixoptimization.

Similarly, from equation (16),

$\begin{matrix}{{\begin{bmatrix}p_{32} \\P_{42}\end{bmatrix} = {- {{{\frac{1}{\Delta}\begin{bmatrix}h_{44} & {- h_{34}} \\{- h_{43}} & h_{33}\end{bmatrix}}\begin{bmatrix}h_{31} & h_{32} \\h_{41} & h_{42}\end{bmatrix}}\begin{bmatrix}p_{12} \\p_{22}\end{bmatrix}}}},{and}} & (21) \\{\begin{bmatrix}c_{12} \\c_{22}\end{bmatrix} = {{A\begin{bmatrix}p_{12} \\p_{22}\end{bmatrix}}.}} & (22)\end{matrix}$

By combining equations (20) and (22), we can establish a relationbetween

$\begin{bmatrix}c_{11} & c_{12} \\c_{21} & c_{22}\end{bmatrix}\mspace{14mu}{{and}\mspace{14mu}\begin{bmatrix}p_{11} & p_{12} \\p_{21} & p_{22}\end{bmatrix}}$as follows:

$\begin{matrix}{{\begin{bmatrix}c_{11} & c_{12} \\c_{21} & c_{22}\end{bmatrix} = {A\begin{bmatrix}p_{11} & p_{12} \\p_{21} & p_{22}\end{bmatrix}}},} & (23)\end{matrix}$and the parameters in

$\left\lbrack \left. \quad\begin{matrix}p_{11} & p_{12} \\p_{21} & p_{22}\end{matrix} \right\rbrack \right.$provide for optimization of the channel matrix

$\begin{bmatrix}c_{11} & c_{12} \\c_{21} & c_{22}\end{bmatrix}.$Note that in the pre-equalization case,

$\left\lbrack \left. \quad\begin{matrix}p_{11} & p_{12} \\p_{21} & p_{22}\end{matrix} \right\rbrack \right.$is selected in such a way that

$\left\lbrack \left. \quad\begin{matrix}c_{11} & c_{12} \\c_{21} & c_{22}\end{matrix} \right\rbrack \right.$is set to

$\begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix}.$On the contrary, in this procedure, two goals are achieved duringpre-coding, namely, separating the layers with respect to receivers andallocating transmitting power to facilitate equal layer performance.Since individual layers now no longer need to be separated at atransmitter, and power allocation is not pre-equalization, the matrix

$\quad\begin{bmatrix}p_{11} & p_{12} \\p_{21} & p_{22}\end{bmatrix}$may be chosen according to different criteria, such as improving channelmatrix condition and providing proportional power allocation, forexample.

To improve channel matrix condition,

${\det\left( \begin{bmatrix}c_{11} & c_{12} \\c_{21} & c_{22}\end{bmatrix} \right)}$should be maximized. One possible way to achieve this is to force theelements in A to add constructively to form the diagonal elements c₁₁and c₂₂. In particular, the pre-coding weights for the UE-1 102 may beset to

$\begin{matrix}\left\{ {\begin{matrix}{p_{11} = {va}_{11}^{*}} \\{p_{21} = {va}_{12}^{*}} \\{p_{12} = {va}_{21}^{*}} \\{p_{22} = {va}_{22}^{*}}\end{matrix},} \right. & (24)\end{matrix}$where v is a power normalization factor and the elements a_(ij) areelements of A. Substituting (24) into (23) yields

$\begin{matrix}{\begin{bmatrix}c_{11} & c_{12} \\c_{21} & c_{22}\end{bmatrix} = {{v\begin{bmatrix}{{a_{11}}^{2} + {a_{12}}^{2}} & {{a_{11}a_{21}^{*}} + {a_{12}a_{22}^{*}}} \\{{a_{21}a_{11}^{*}} + {a_{22}a_{12}^{*}}} & {{a_{21}}^{2} + {a_{22}}^{2}}\end{bmatrix}}.}} & (25)\end{matrix}$

It can thus be seen that c₁₁ and c₂₂ are enhanced constructively, whilec₁₂ and c₂₁ are constructed randomly. Therefore, the condition of Cbecomes more robust. From a beam-forming point of view, layer-1 s₁ ⁽¹⁾and layer-2 s₂ ⁽¹⁾ are beamed onto antenna-i 108 and antenna-2 110,respectively, following the MRC (Maximum Ratio Combining) criterion. Theelements c₁₂ and c₂₁ represent both inter-layer interference andreceiver diversity. Recall that λ₁+λ₂=c₁₁+c₂₂, where λ_(i) (i=1,2) areeigenvalues of

$\begin{bmatrix}c_{11} & c_{12} \\c_{21} & c_{22}\end{bmatrix}.$Since c₁₁ and c₂₂ are enhanced constructively, so are λ₁+λ₂. Inaddition, according to Cauchy-Schwarz Inequality, we always have

${{\det\left( \begin{bmatrix}c_{11} & c_{12} \\c_{21} & c_{22}\end{bmatrix} \right)}} > 0$when

$\left\{ {\begin{matrix}{a_{11} \neq a_{21}} \\{a_{12} \neq a_{22}}\end{matrix}.} \right.$

For the UE-2 104, the elements of P may be selected in an analogousmanner. The elements p₁₃, p₂₃, p₁₄, and p₂₄ are preferably selected toforce c₃₁=c₃₂=c₄₁=c₄₂=0, and p₃₃, p₄₃, p₃₄, and p₄₄ are preferablyselected as p₃₃=va₃₃*, p₄₃=va₃₄*, p₃₄=va₄₃*, and p₄₄=va₄₄*, where v isas defined above. For UE-2 104, however,

${A = {\begin{bmatrix}h_{33} & h_{34} \\h_{43} & h_{44}\end{bmatrix} - {{{\frac{1}{\Delta}\begin{bmatrix}h_{31} & h_{32} \\h_{41} & h_{42}\end{bmatrix}}\begin{bmatrix}h_{22} & {- h_{12}} \\{- h_{21}} & h_{11}\end{bmatrix}}\begin{bmatrix}h_{13} & h_{14} \\h_{14} & h_{24}\end{bmatrix}}}},$and Δ=h₁₁h₂₂−h₁₂h₂₁.

A new scheme to further enhance multi-user MIMO system performance hasbeen described. The above embodiments are based on splitting theinterference cancellation task between a transmitter and receivers.Specifically, a transmitter performs inter-user separation pre-coding todefine sub-MIMO channels, while receivers perform individual layerseparation. The transmitter and each receiver benefit from this taskpartitioning. From the transmitter point of view, since it is no longerrequired to provide individual layer separation, it has the freedom toperform beamforming, which results in more robust equivalent channelmatrix, and proportional power allocation. At the receiver, sincemultiple antennas are receiving signals from multiple layers, when theML decoding algorithm is used, additional diversity gain can beachieved.

The above description relates primarily to MIMO systems in whichsub-MIMO channels between the BTS 100 and each UE 102, 104 have the samedimension. However, it should be appreciated that these embodiments ofthe invention may also be extended to systems in which UEs do not havethe same number of antennas, such that sub-MIMO channels of differentdimensions are supported in the same system.

Further embodiments of the present invention will be best appreciated inconjunction with the following detailed analysis of MIMO, particularlyMIMO BLAST. FIG. 6 is a block diagram of a MIMO BLAST system.

The system of FIG. 6 includes a BTS 130 and a UE 132. The BTS includes amodulator 134, an S/P (serial-to-parallel) converter 136, and aplurality of M antennas 140. At the UE 132, communication signalsreceived at a plurality of N antennas 142 are decoded in a decoder 150to recover the transmitted signals s₁ ⁽¹⁾, . . . , s_(M) ⁽¹⁾ at 152.Communication channel noise is represented by the adders 144, 146, 148.

FIG. 6 shows a general case of an M×N MIMO system. Although threerepresentative signals and antennas are shown at the BTS 130 and the UE132, the invention is in no way restricted any particular dimension ofMIMO system. M and N may be equal to, larger than, or smaller than 3.Also, as described above in conjunction with FIG. 1, BTSs and UEstypically include further components that have not been shown in FIG. 6to avoid congestion in the drawing.

According to the basic point-to-point Layered STC architecture known asBLAST, a sequence of the modulated symbols, each symbol having aduration of T, from the modulator 134, is serial-to-parallel convertedin the S/P converter 136 into parallel transmission blocks in thesignals 138. Each transmission block consists of K_(ch) symbols, whereK_(ch) is the number of spatial channels. In the downlink transmissioncase, K_(ch) is equal to the number of antennas, M, at the BTS 130. Allof the symbols of a transmission block are simultaneously radiated intospace, and each symbol is radiated by a respective one of the antennas140. As the equivalent time duration of the transmission block relativeto the original modulated signal output by the modulator 134 is K_(ch)T,the radiated signal requires spectrum width equal to a fraction(1/K_(ch)) of that of the original signal. This achieves a very highspectral efficiency.

The reception of a signal in such systems requires multiple antennas,with N≧M. The model of a received signal is described by followingvector-matrix expressiony =√{square root over (P _(s)/M)}H s + η,   (26)where

-   -   P_(s) is the total transmitted power; and    -   y, M, H, s, and η are as defined above.

The elements of H are preferably independent Gaussian complex randomvariables, with zero mean and E{|h_(mn)|²}=1 variances. η alsopreferably has zero mean, and R=2σ_(η) ²I_(N) covariance matrix, whereI_(N) is an N×N-dimensioned unit matrix and σ_(η) ² is the variance ofone quadrature component of η. The mean power of each radiated symbol isequal to unity, i.e., E{|s_(m) ²}=1.

The solution for linear estimation of a vector of modulated symbols canbe carried out, for example, by the ZF criterion expressed as{tilde over (s)}=H ⁺ y /(P _(s) /M).   (27)

Estimation by the MMSE criterion is as follows:{tilde over (s)} =(H′H/N+(2σ_(η) ² /P _(s))I _(M))⁻¹(H′√{square rootover (M)}) y =(H′√{square root over (M)})(HH′/N+(2σ_(η) ² /P _(s))I_(M))⁻¹ y,   (28)where I_(M) is an M×M-dimensioned unit matrix

The MMSE algorithm provides a significant gain as contrasted with the ZFalgorithm in a channel with Rayleigh fading.

For single user point-to-point MIMO with an open-loop transmission, theMIMO channel capacity is proportional to min{N,M}. It well known thatthe throughput capacity of MIMO grows linearly with an increase ofmin{N,M}. Conventional MIMO-BLAST receiver processing schemes requireco-processing of signals received by all antennas in the MIMO system.However, such a method cannot be applied for multi-user combined MIMOsystems, as the signals received by all other UEs may not always beaccessible to each UE. Therefore, usage of MIMO BLAST in multi-usersystems to provide for multiple access may be inefficient, despite amarginal increase of throughput capacity at the expense of an increasein the number transmitting and receiving antennas.

In the multi-user point-to-multi-point case, since the inter-usercommunication is typically not done, the capacity of a multi-user systemwith open-loop transmit diversity is determined by min{M,N₁,N₂, . . .N_(U)}, where U, as above, is the number of UEs in a multi-user system.Thus, common throughput capacity will be limited by UEs with the leastnumber of receiving antennas.

FIG. 7 is a block diagram of a multi-user MIMO system in accordance withanother embodiment of the invention. The system of FIG. 7 includes a BTS160 with a beamforming module 166 and a plurality of antennas 168, 170,172, 174, and two UEs 162, 164, each having a pair of antennas 176/178,182/184 and a decoder 180, 186. The beamforming module 166 includesrespective beamformers 188, 190 and signal combiners 192/194, 196/198for each UE 162, 164. The beamformers 188, 190, and possibly the signalcombiners 192, 194, 196, 198, are preferably implemented using signalprocessing software in conjunction with either a DSP or ageneral-purpose processor at the BTS 160. The decoders 180, 186 are alsopreferably software-based.

Closed-Loop Transmit Diversity (CLTD), with different configurations forthe two UEs 162, 164, is provided by feeding back channel stateinformation shown as H₁, H₂ from the UEs 162, 164 to the BTS 160. Itshould be appreciated that the combined 4×2 (M=4, N=2, U=2) multi-usersystem of FIG. 7 is one illustrative example of a MIMO system to whichthe present invention may be applied, and that the invention is notlimited thereto. This will become apparent from the followinggeneralized analysis of multi-user MIMO with CLTD.

The BTS 160 has K_(ch) spatial channels and M=4 antennas 168, 170, 172,174, with K_(ch)≦M. In a multi-user system, there exist only K_(ch,i)channels to the i^(th) UE. Signal vectors s(i) of symbols intended forthe i th UE are multiplied in the beamformers 188, 190 by respectiveelements of F=[F⁽¹⁾,F⁽²⁾, . . . F^((U))], which is a spatial codingmatrix of spatial coding weights. Each element F^((i)) is a personalbeamforming matrix for i^(th) UE with M×K_(ch,i) dimension andsatisfyingtr{F ^((i)) F ^((i)) ^(T) }=tr{F ^((i)) ^(T) F ^((i)) }=P _(s) , i=1, 2,. . . , U   (29)where

-   -   tr{●} is the trace of a matrix.

As shown, signals output from each beamformer 188, 190 are combined inthe signal combiners 192, 194, 196, 198 and output to respectiveantennas 168, 170, 172, 174.

In this case, a constant and equal transmitted power is applied tosignals for all UEs. The vector signal received by the i^(th) UE isdescribed by

$\begin{matrix}{{{\overset{\rightharpoonup}{y}}^{(i)} = {{{H^{(i)}F^{(i)}{\overset{\rightharpoonup}{s}}^{(i)}} + {H^{(i)}{\sum\limits_{{n = 1},{n \neq i}}^{U}\;{F^{(n)}{\overset{\rightharpoonup}{s}}^{(n)}}}} + \eta^{(i)}} = {{H^{(i)}F\overset{\rightharpoonup}{s}} + {\overset{\rightharpoonup}{\eta}}^{(i)}}}}{where}} & (30)\end{matrix}$

-   -   s ^((i))=[s₁ ^((i)),s₂ ^((i)) . . . s_(K) _(ch,i) ^((i))]^(T) is        a K_(ch,i)-dimensioned vector of symbols of the i^(th) UE;    -   s=[s₁,s₂ . . . s_(M)]^(T) is a K_(ch)-dimensioned vector of        symbols of all UEs;

$K_{ch} = {\sum\limits_{i = 1}^{U}\; K_{{ch},i}}$is the total number of channels for the BTS;

-   -   y ^((i))=[y₁ ^((i)),y₂ ^((i)) . . . y_(N) ^((i))]^(T) is an        N-dimensioned complex vector of signals received at the i^(th)        UE;    -   H^((i)) is an N×M-dimensioned matrix of complex channel gains        from the BTS to the i^(th) user;    -   η^((i))=[η₁ ^((i)),η₂ ^((i)) . . . η_(N) ^((i))]^(T) is an        N-dimensioned complex vector of noise of observation for the        i^(th) user with zero mean and R^((i))=2σ_(η) ^((i)) ² I_(N)        covariance matrix.

In multi-user systems in which UEs have different numbers of antennas, Nwould be replaced with N_(i) above, where N_(i) is the number ofantennas at the i^(th) UE.

CLTD multi-user MIMO can be considered as the optimization of signaldetection matrices. When the number of transmitting antennas is the sameas the number of receiving antennas, so-called transmit and receivechannel reciprocity exists. In this case, optimum weighting coefficientsfor beamforming for each UE are calculated at a BTS. The computedweighting coefficients are then used for radiation of a signal by theBTS. If the total number of receiving antennas of all UEs is equal tothe number of transmitting antennas of the BTS, the full division ofeach transmitted signal, directly, on UE receiving antennas issatisfied.

Now consider a reverse channel problem for a single user, to determine aCLTD matrix for data transmission from an i^(th) UE, with N_(i) antennason K_(ch,i) parallel channels. The virtual reverse channel signal,observed at the BTS 166, is described byŷ ^((i)) =Ĥ ^((i)) {circumflex over (F)} ^((i)) ŝ ^((i))+ {circumflexover (η)} ^((i))   (31)where

-   -   Ĥ^((i))=[H^((i))] is an M×N_(i)-dimensional channel matrix of        the virtual reverse MIMO channel;    -   {circumflex over (F)}^((i)) is the optimal virtual CLTD space        time coding matrix of N_(i)×K_(ch,i) dimension; and    -   the “^” symbol indicates matrices and vectors associated with        the virtual reverse channel.

There are several ways to construct the {circumflex over (F)}^((i))matrix. Consider first a singular decomposition of the channel matrix H,{tilde over (H)}=UΛV^(H), where U and V are unitary matrices with M×Mand N_(i)×N_(i) dimensions, respectively, and Λ is a non-negativediagonal matrix with M×N_(i) dimension. The squares of diagonal elementsof Λ are equal to eigenvalues of the ĤĤ′ matrix. The columns of U areeigenvectors of the ĤĤ′ matrix, and the columns of V are alsoeigenvectors of the ĤĤ′ matrix.

The optimum value of {circumflex over (F)} can be shown to be{circumflex over (F)}= VΦ,   (32)where V is a matrix with N_(i)×K_(ch,i) dimension, constructed fromK_(ch,i) columns of V, and Φ is a diagonal matrix with K_(ch,i)×K_(ch,i)dimension having non-negative diagonal elements satisfying the followingcondition:

$\begin{matrix}{{{tr}\left( {\hat{F}\hat{F^{\prime}}} \right)} = {{\sum\limits_{k = 1}^{K_{{ch},i}}\;{\phi_{k,k}}^{2}} = {P_{s}.}}} & (33)\end{matrix}$The diagonal elements of matrix Φ determine channel power allocation. Auniform power allocation givesφ_(k,k) ² =P _(s) /K _(ch,i).   (34)

Some possible alternative versions of power allocation include:

-   1. MMSE criterion

$\begin{matrix}{{{\phi_{k,k}}^{2} = {2{{\sigma_{\eta}}^{2}\left\lbrack {\frac{\mu}{\sqrt{\xi_{k,k}}} - \frac{1}{\xi_{k,k}}} \right\rbrack}^{+}}};} & (35)\end{matrix}$

-   2. Minimum Symbol-Error-Rate (MSER) criterion

$\begin{matrix}{{{\phi_{k,k}}^{2} = {\frac{2{\sigma_{\eta}}^{2}}{\xi_{k,k}}\left\lbrack {{\log\left( \frac{\xi_{k,k}}{2{\sigma_{\eta}}^{2}} \right)} - \mu} \right\rbrack}^{+}};{and}} & (36)\end{matrix}$

-   3. Maximum Capacity and Information Rate (MCIR) criterion, also    commonly known as the water-filling algorithm

$\begin{matrix}{{{\phi_{k,k}}^{2} = \left( {\mu - \frac{2{\sigma_{\eta}}^{2}}{\xi_{k,k}}} \right)^{+}},} & (37)\end{matrix}$where

${(\bullet)^{+} = {{\max\left( {\bullet,0} \right)} = {\frac{1}{2}\left( {{\bullet } + \bullet} \right)}}};$

ξ_(k,k)=λ_(k,k) ² are eigenvalues of the ĤĤ′ matrix, and λ_(k,k) arediagonal elements of the Λ matrix; and

-   -   μ is a factor that is selected to define each criteria.

After the CLTD matrix is constructed at the transmitter, equation (31)becomesŷ ^((i)) =Ĥ _(F) ^((i)) ŝ ^((i))+{circumflex over (η)}^((i)) i=1,2, . .. U,   (38)where

-   -   Ĥ_(F) ^((i))=(Ĥ^((i)){circumflex over (F)}^((i)))/√{square root        over (2σ_(η,) _(i) ²)} is a matrix of the virtual reverse MIMO        channel with M×K_(ch,i) dimension; and    -   η=        (0,I_(M)) is an M-dimensioned complex vector of a virtual noise        observed at a BTS, with zero mean and R _(η)=I_(M) covariance        matrix.

A personal beamforming matrix {circumflex over (F)}^((i)) can thereby beconstructed in a closed loop fashion for each individual user in theabsence of other users. However, in a multi-user scenario, the presenceof inter-user MIMO interference prevents such a straightforwarduser-specific personal beamforming approach. An embodiment of theinvention provides a solution for the multi-user CLTD by using the MMSEcriterion to minimize the inter-user MIMO interference, and a networksolution for optimizing the multi-user MIMO allocation.

To integrate signals of all users into a virtual model, it is possibleto re-write the virtual reverse MIMO channel model for multiple users,as given below:ŷ=Ĥ _(F) ŝ+ {circumflex over (η)},   (39)where

-   -   ŝ=[( ŝ ⁽¹⁾)^(T),( ŝ ⁽²⁾)^(T), . . . ( ŝ        ^(({circumflex over (K)}) ^(ch) ⁾)^(T)]^(T) is a {circumflex        over (K)}_(ch)-dimensioned vector of symbols, and

${{\hat{K}}_{ch} = {\sum\limits_{i = 1}^{U}\;{\hat{K}}_{{ch},i}}};$and

-   -   Ĥ_(F)=└Ĥ_(F) ⁽¹⁾,Ĥ_(F) ⁽²⁾, . . . Ĥ_(F) ^({circumflex over (K)})        ^(ch) ┘ is an M×{circumflex over (K)}_(ch)-dimensioned matrix of        the reverse virtual MIMO Channel.

It should be noted that {circumflex over (K)}_(ch)≦K_(ch)≦M, i.e. thenumber of estimated symbols is less than or equal to the number ofreceived signals. In this situation, a very effective estimation can becarried out using, for example, a linear MMSE algorithm, as follows:

$\begin{matrix}{{{\hat{\overset{\rightharpoonup}{s}} = {\hat{G}\hat{\overset{\rightharpoonup}{y}}}},{where}}{\hat{G} = {{\left( {{{{\hat{H}}_{F}}^{\prime}{\hat{H}}_{F}} + I_{{\hat{K}}_{{ch},i}}} \right)^{- 1}{{\hat{H}}_{F}}^{\prime}} = {{{\hat{H}}_{F}}^{\prime}\left( {{{\hat{H}}_{F}{{\hat{H}}_{F}}^{\prime}} + I_{{\hat{K}}_{{ch},i}}} \right)}^{- 1}}}{or}} & (40) \\{{{{\hat{\overset{\rightharpoonup}{s}}}^{(i)} = {{\hat{G}}^{(i)}\hat{\overset{\rightharpoonup}{y}}}},{where}}{{{\hat{G}}^{(i)} = {{{{\hat{H}}_{F}}^{(i)}}^{\prime}\left( {{{\hat{H}}_{F}{{\hat{H}}_{F}}^{\prime}} + I_{{\hat{K}}_{{ch},i}}} \right)}^{- 1}},{i = 1},2,{{\ldots\mspace{14mu} U};{and}}}{\hat{G} = {\begin{bmatrix}{\hat{G}}^{(1)} \\{\hat{G}}^{(2)} \\\vdots \\{\hat{G}}^{(U)}\end{bmatrix}.}}} & (41)\end{matrix}$

Using a principle of identity (or duality) of receiving and transmittingchannels, an optimum demodulation matrix Ĝ is preferably used forgenerating F=[F⁽¹⁾,F⁽²⁾, . . . F^((U))], the CLTD or beamforming matrix,at a BTS. In this case, personal beamforming matrices are preferablydetermined by

$\begin{matrix}{{F^{(i)} = {{\sqrt{P_{s}}\frac{{\hat{G}}^{{(i)}^{\prime}}}{\sqrt{{tr}\left( {{\hat{G}}^{{(i)}^{\prime}}{\hat{G}}^{(i)}} \right)}}} = {\sqrt{P_{s}}\frac{{\hat{G}}^{{(i)}^{\prime}}}{\sqrt{\sum\limits_{m = 1}^{K_{{ch},i}}\;{\sum\limits_{n = 1}^{U}\;{{\hat{g}}_{m,n}^{(i)}}^{2}}}}}}},} & (42)\end{matrix}$where ĝ_(m,n) is (m,n)^(th) element of Ĝ^((i)).

The model of an observed (received) signal at the input of the UE of thei^(th) user can be written as

$\begin{matrix}\begin{matrix}{{\overset{\rightharpoonup}{y}}^{(i)} = {{H^{(i)}F^{(i)}{\overset{\rightharpoonup}{s}}^{(i)}} + {H^{(i)}{\sum\limits_{{n = 1},{n \neq i}}^{U}\;{F^{(n)}{\overset{\rightharpoonup}{s}}^{(n)}}}} + {\overset{\rightharpoonup}{\eta}}^{(i)}}} \\{{= {{H_{F}^{({i,i})}{\overset{\rightharpoonup}{s}}^{(i)}} + {H^{(i)}{\sum\limits_{{n = 1},{n \neq i}}^{U}\;{H_{F}^{({i,n})}{\overset{\rightharpoonup}{s}}^{(i)}}}} + {\overset{\rightharpoonup}{\eta}}^{(i)}}}\mspace{14mu},} \\{= {{H_{F}^{(i)}\overset{\rightharpoonup}{s}} + {\overset{\rightharpoonup}{\eta}}^{(i)}}}\end{matrix} & (43)\end{matrix}$where H_(F) ^((i,n))=H^((i))F^((n)), H_(F) ^((i))=[H_(F) ^((i,l)), . . .H_(F) ^((i,U))]. In some communication systems, the BTS can measure theH_(F) ^((i,n)) n=1,2 . . . U matrices. In other systems, the UEs feedback channel matrices to the BTS.

The BTS can compute the personalized demodulation matrix for the i^(th)user as

$\begin{matrix}{\begin{matrix}{{\hat{G}}^{(i)} = {H_{F}^{{({i,i})}^{\prime}}\left\lbrack {{\sum\limits_{n = 1}^{U}\;{{\hat{H}}_{F}^{({i,n})}{\hat{H}}_{F}^{{({i,n})}^{\prime}}}} + {2\sigma_{\eta}^{2}I_{N_{i}}}} \right\rbrack}^{- 1}} \\{{{= {H_{F}^{{({i,i})}^{\prime}}\left\lbrack {{{\hat{H}}_{F}^{(i)}{\hat{H}}_{F}^{{(i)}^{\prime}}} + {2\sigma_{\eta}^{2}I_{N_{i}}}} \right\rbrack}^{- 1}}\mspace{20mu},}\mspace{31mu}}\end{matrix}\mspace{11mu}} & (44)\end{matrix}$and sends to each UE its respective demodulation matrix Ĝ^((i)).

Integrating all transformations for calculating the CLTD matrix(personal beamforming matrices F^((i)) at a BTS side and personalbeamforming matrices Ĝ^((i)) at UE side), yields the following algorithmto achieve CLTD based multi-user MIMO transmission in accordance with anembodiment of the invention:

At a BTS Side:

$\begin{matrix}\begin{matrix}{F^{(i)} = {\sqrt{P_{s}}\frac{{\hat{G}}^{{(i)}^{\prime}}}{\sqrt{{tr}\left( {{\hat{G}}^{{(i)}^{\prime}}{\hat{G}}^{(i)}} \right)}}}} \\{{\hat{G}}^{(i)} = {{\hat{H}}_{F}^{{(i)}^{\prime}}\left\lbrack {{{\hat{H}}_{F}{\hat{H}}_{F}^{\;^{\prime}}} + I_{N_{i}}} \right\rbrack}^{- 1}} \\{{\hat{H}}_{F} = \left\lbrack {{\hat{H}}_{F}^{(1)},\;{\ldots\mspace{14mu}{\hat{H}}_{F}^{(U)}}} \right\rbrack} \\{{\hat{H}}_{F}^{(i)} = {\left( {{\hat{H}}^{(i)}{\hat{F}}^{(i)}} \right)/\sqrt{2\sigma_{\eta,i}^{2}}}} \\{{\hat{H}}^{(i)} = \left\lbrack H^{(i)} \right\rbrack^{\prime}} \\{{\hat{F}}^{(i)} = {{\overset{\_}{V}}^{(i)}\Phi^{(i)}}}\end{matrix} & (45)\end{matrix}$At a UE Side:

$\begin{matrix}\begin{matrix}{{\hat{G}}^{(i)} = {H_{F}^{{({i,i})}^{\prime}}\left\lbrack {{\sum\limits_{n = 1}^{U}\;{{\hat{H}}_{F}^{({i,n})}{\hat{H}}_{F}^{{({i,n})}^{\prime}}}} + {2\sigma_{\eta}^{2}I_{N_{i}}}} \right\rbrack}^{- 1}} \\{H_{F}^{({i,n})} = {H^{(i)}F^{(n)}}}\end{matrix} & (46)\end{matrix}$

In the above multi-user system, the following constraints are preferablysatisfied:

$\begin{matrix}{\begin{matrix}{{\sum\limits_{i = 1}^{U}\; K_{{ch},i}} \leq M} \\{K_{{ch},i} \leq N_{i}}\end{matrix}.} & (47)\end{matrix}$

The first constraint of (47) specifies that the total number of parallelchannels should not exceed the number of BTS antennas, which actuallydetermines the number of parallel spatial channels. The secondconstraint specifies that the number of receiving antennas at a UEshould be greater than or equal to the number of parallel spatialchannels assigned to it. These conditions allow the use of linearmethods to construct the personal beamforming matrices at both the BTSand UE for all users.

For TDD (Time Division Duplexing) communications, all the computing canbe done at a BTS. The BTS determines all relevant parameters, calculatesboth the BTS and UE beamforming matrices, and feeds back a personalbeamforming receive matrix Ĝ^((i)) to each UE.

For the FDD (Frequency Division Duplexing) case, all the UEs candetermine and feed back the initial SVD (Singular Value Decomposition)beamforming matrix {circumflex over (F)}^((i)) to the BTS, which thenjointly integrates all {circumflex over (F)}^((i)) matrices to computethe dedicated beamforming transmit matrix F^((i)) for each UE. The BTSthen computes and sends a respective beamforming receive matrix Ĝ^((i))to each mobile terminal.

Embodiments of the above CLTD-based multi-user MIMO exhibit performanceadvantages relative to conventional open-loop solutions. For thepurposes of comparison, the following simulation conditions were used:(1) R=½ turbo coding, block length 1280 bits, (2) QPSK (Quadrature PhaseShift Keying) modulation, (3) MMSE receiver for all schemes, and (4)ideal channel feedback.

Before proceeding with a discussion of simulation results, each of theconventional technologies with which comparison is made will be brieflydescribed.

Null beamforming is an open-loop scheme where the number of transmittingantennas is equal to the total number of receiving antennas, namely

${{\sum\limits_{i = 1}^{U}\; K_{{ch},i}} = M},$and K_(ch,i)=N_(i), such as shown in FIG. 8. Therefore, it is possibleto perform null beamforming to reduce the inter-user interference of theBTS antenna emissions to the other users. The configuration of 2×8×2 fora 4 user environment was considered in the simulations presented below.

In one known open-loop multi-user BLAST technique, the number oftransmitting antennas is the same as the number of receiving antennasfor each UE, as shown in FIG. 9, and the number of channels is equal tothe number of transmitting antennas. A UE receive signal can beexpressed as

${{\overset{\rightharpoonup}{y}}^{(i)} = {{\sqrt{\frac{P_{s}}{K_{{ch},i}}}H^{(i)}\overset{\rightharpoonup}{s}} + {\overset{\rightharpoonup}{\eta}}^{(i)}}},$where s is an M-dimensioned vector of symbols. At N_(i)≧M, it ispossible to use an MMSE algorithm for demodulation of the entire svector. Therefore, from the demodulated s vector, the symbolstransmitted to a particular user are extracted. This scheme is generallyknown as Multi-user BLAST. We evaluate the configuration for such asystem with dimension 2×8×8 and U=4.

Let K_(ch,i)=M/U and N_(i)≧M. In this case, it is possible for a BTS togroup together the signals of each user into units with M symbols tothereby obtain units to sequentially transmit in a round robin fashion.At a UE side, the i^(th) time slot is demodulated by the i^(th) userwith an MMSE decoder. This type of system in shown in FIG. 10. Theperformance of such a system is the same as a point-to-point BLASTsystem, with the compression of the duration of radiation of a signal ofeach user by a factor of 1/U in time without increasing the transmitpower. We evaluate the configuration 2×8×8 and U=4 for such a system.

FIGS. 11-13 show simulation results for embodiments of the invention andthe above known technologies.

FIG. 11 shows a plot of BLER versus SNR for an embodiment of theinvention and several known communication schemes. It can be seen fromFIG. 11 that for 8 transmit antennas at a BTS and 4 concurrent userseach with 2 receive antennas at each UE, a CLTD scheme according to anembodiment of the invention has 19 dB gain over open-loop nullbeamforming. This simulated CLTD scheme also achieves virtually the sameperformance as the 4 users open-loop multi-user BLAST each with 8receive antennas. In this case, a reduction of 6 receive antennas ateach UE can be realized. Given 8 receive antennas at the UE side, theproposed CLTD scheme achieves 5 dB gain over the conventional open-loopTDM BLAST, 9 dB gain over multi-user BLAST, and 3.8 dB gain over theclosed-loop TDM BLAST.

FIG. 12 is a plot of BLER versus Eb/No for an embodiment of theinvention and several known communication schemes. The simulated CLTDscheme from which the plot of FIG. 12 was generated is clearly superiorto point-to-point MIMO (open-loop BLAST) and null beamforming.

FIG. 13 is a plot of BLER versus SNR for several embodiments of theinvention, and demonstrates the scalability of multi-user MIMO and MISO(Multiple Input Single Output) configurations according to embodimentsof the invention. For the different configurations indicated in FIG. 13,all users achieve substantially the same level of QoS (Quality ofService).

Numerous modifications and variations of the present invention arepossible in light of the above teachings. It is therefore to beunderstood that within the scope of the appended claims, the inventionmay be practised otherwise than as specifically described herein.

Of course it is to be understood that in a given application, specificparameters may change. For example, different numbers of users, transmitantennas, and receive antennas may change the particular derivationdetails and equations above. However, adaptation of the above andfurther embodiments of the invention to other types and dimensions ofsystems than those explicitly described will be apparent to thoseskilled in the art from the foregoing.

It should also be appreciated that references to transmitting or sendingsignals is not intended to limit the invention only to embodiments inwhich signals are transmitted exactly as generated, without any furtherprocessing. For example, signals may be compressed or otherwiseprocessed prior to transmission, or stored for subsequent transmissionat a later time.

We claim:
 1. A method of processing signals to be transmitted toreceivers on a plurality of communication channels, comprising:determining pre-coding signal weights based on channel state informationassociated with the plurality of communication channels to provideproportional power allocation to the signals; and applying thepre-coding signal weights to the signals, wherein the pre-coding signalweights are elements of a pre-coding matrix P, wherein determiningfurther comprises determining the pre-coding signal weights to enhancediagonal elements of a combined communication channel matrix C=HP, whereH is a matrix of the channel state information; and wherein${P = \begin{bmatrix}p_{11} & p_{12} \\p_{21} & p_{22}\end{bmatrix}},$  wherein ${H = \begin{bmatrix}h_{11} & h_{12} \\h_{21} & h_{22}\end{bmatrix}},$  and wherein determining comprises selecting thepre-coding signal weights of P such that${{p_{11}} \propto {{h_{11} - \frac{h_{12}h_{21}}{h_{22}}}}};$${{p_{22}} \propto {{h_{22} - \frac{h_{12}h_{21}}{h_{11}}}}};$${p_{12} = {- \frac{h_{12}p_{22}}{h_{11}}}};{and}$$p_{21} = {- {\frac{h_{21}p_{11}}{h_{22}}.}}$
 2. A method of processingsignals transmitted to receivers on a plurality of communicationchannels, comprising: at a transmitter: determining pre-coding signalweights based on channel state information associated with the pluralityof communication channels to provide proportional power allocation tothe signals; applying the pre-coding signal weights to the signals; andtransmitting weighted signals to the receivers on the plurality ofcommunication channels; wherein the transmitter comprises four antennascomprising two sub-groups of antennas comprising two antennas each, eachsub-group of two antennas respectively associated with two sub-groups ofcommunication channels of the plurality of communication channels, eachsub-group of communication channels comprising two communicationchannels; and at each of the receivers: receiving a subset of theweighted signals over one of the sub-groups of communication channels;and decoding the subset of the weighted signals using inverses of thepre-coding signal weights based on the channel state informationassociated with the one of the sub-groups of communication channels;wherein the pre-coding signal weights are elements of a pre-codingmatrix P, and wherein determining further comprises determining thepre-coding signal weights to enhance diagonal elements of a combinedcommunication channel matrix C=HP, where H is a matrix of the channelstate information; and wherein determining the pre-coding signal weightscomprises selecting the pre-coding signal weights of P such that${C = {{HP} = \begin{bmatrix}c_{11} & c_{12} & 0 & 0 \\c_{21} & c_{22} & 0 & 0 \\0 & 0 & c_{33} & c_{34} \\0 & 0 & c_{43} & c_{44}\end{bmatrix}}},$ where a group of the first two rows of C is associatedwith a first of the two sub-groups of communication channels, a group ofthe third and fourth rows of C is associated with a second of the twosub-groups of communication channels, a group of the first two columnsof C is associated with a first of the two sub-groups of two antennas,and a group of the third and fourth columns of C is associated with asecond of the two sub-groups of two antennas.
 3. A method of processingsignals to be transmitted to receivers on a plurality of communicationchannels, comprising: determining pre-coding signal weights based onchannel state information associated with the plurality of communicationchannels to provide proportional power allocation to the signals; andapplying the pre-coding signal weights to the signals, wherein thesignals comprise respective groups of signals to be transmitted to thereceivers, wherein determining the pre-coding signal weights furthercomprises determining the pre-coding signal weights to separate therespective groups of signals, wherein the method is implemented at atransmitter in a multi-user MIMO (Multiple Input Multiple Output)communication system that provides respective N ×N sub-MIMO channelsfrom the transmitter to the receivers, wherein each of the groups ofsignals comprises N signals, and wherein determining the pre-codingsignal weights further comprises determining elements of a pre-codingmatrix P such that a combined communication channel matrix C =HP has aform of U N ×N sub-matrices, diagonal elements of which are respectivediagonal elements of C, and elements of C outside the N ×N sub-matricesare forced to zero.
 4. The method of claim 3, wherein the transmitterhas M=4 antennas, wherein U=2, N=2, ${P = \begin{bmatrix}p_{11} & p_{12} & p_{13} & p_{14} \\p_{21} & p_{22} & p_{23} & p_{24} \\p_{31} & p_{32} & p_{33} & p_{34} \\p_{41} & p_{42} & p_{43} & p_{44}\end{bmatrix}},{H = \begin{bmatrix}h_{11} & h_{12} & h_{13} & h_{14} \\h_{21} & h_{22} & h_{23} & h_{24} \\h_{31} & h_{32} & h_{33} & h_{34} \\h_{41} & h_{42} & h_{43} & h_{44}\end{bmatrix}},{C = {{HP} = \begin{bmatrix}c_{11} & c_{12} & 0 & 0 \\c_{21} & c_{22} & 0 & 0 \\0 & 0 & c_{33} & c_{34} \\0 & 0 & c_{43} & c_{44}\end{bmatrix}}},$ wherein determining elements of P comprises: selectingp₃₁, p₄₁, p₃₂, and p₄₂ to force c₁₃=c₁₄=c₂₃=c₂₄=0; selecting$\left\{ {\begin{matrix}{p_{11} = {va}_{11}^{*}} \\{p_{21} = {va}_{12}^{*}} \\{p_{12} = {va}_{21}^{*}} \\{p_{22} = {va}_{22}^{*}}\end{matrix},} \right.$  where v is a power normalization factor anda_(ij) are elements of A, where $A = {\begin{bmatrix}h_{11} & h_{12} \\h_{21} & h_{22}\end{bmatrix} - {{{\frac{1}{\Delta}\begin{bmatrix}h_{13} & h_{14} \\h_{23} & h_{24}\end{bmatrix}}\begin{bmatrix}h_{44} & {- h_{34}} \\{- h_{43}} & h_{33}\end{bmatrix}}\begin{bmatrix}h_{31} & h_{32} \\h_{41} & h_{42}\end{bmatrix}}}$  and Δ=h₃₃h₄₄−h₃₄h₄₃; selecting p₁₃, p₂₃, p₁₄, and p₂₄to force c₃₁=c₃₂=c₄₁=c₄₂=0; and selecting $\left\{ {\begin{matrix}{p_{33} = {va}_{11}^{*}} \\{p_{43} = {va}_{12}^{*}} \\{p_{34} = {va}_{21}^{*}} \\{p_{44} = {va}_{22}^{*}}\end{matrix},} \right.$  where a_(ij) are elements of A, where${A = {\begin{bmatrix}h_{33} & h_{34} \\h_{43} & h_{44}\end{bmatrix} - {{{\frac{1}{\Delta}\begin{bmatrix}h_{31} & h_{32} \\h_{41} & h_{42}\end{bmatrix}}\begin{bmatrix}h_{22} & {- h_{12}} \\{- h_{21}} & h_{11}\end{bmatrix}}\begin{bmatrix}h_{13} & h_{14} \\h_{14} & h_{24}\end{bmatrix}}}},$  and Δ=h₁₁h₂₂−h₁₂h₂₁.
 5. A method, implemented in aMIMO (Multiple Input Multiple Output) communication system, ofprocessing signals to be concurrently transmitted to receivers over aplurality of communication channels comprising: determining channelstate information for the plurality of communication channels;determining a spatial coding matrix comprising a respective set ofspatial coding weights for each of the receivers based on the channelstate information; applying the respective sets of spatial codingweights in the spatial coding matrix to the signals, wherein the signalscomprise a plurality of groups of at least one signal to be transmittedto respective ones of the receivers; and transmitting the signals to thereceivers, wherein the spatial coding matrix F comprises elements [F⁽¹⁾,F⁽²⁾, . . . F^((U))], U is an integer, and each element F^((i)) is therespective set of spatial coding weights for an i^(th) one of thereceivers and satisfies tr{F(^((i))F^((i)′)}=tr{F^((i)′)F^((i))}=P_(s),i=1,2, . . . , U, where tr{•} is a trace of a matrix, and P_(s) is atotal transmitted power of the signals; wherein determining the spatialcoding matrix comprises determining the elements F^((i)) of F as${F^{(i)} = {\sqrt{P_{s}}\frac{{{\hat{G}}^{(i)}}^{\prime}}{\sqrt{{tr}\left( {{{\hat{G}}^{(i)}}^{\prime}{\hat{G}}^{(i)}} \right)}}}},$where Ĝ^((i))=Ĥ_(F) ^((i)′)(Ĥ_(F)Ĥ′_(F) +I_(N) _(i) )⁻¹, i=1,2, . . . U,is a set of demodulation weights corresponding to F^((i)); Ĥ_(F)=[Ĥ_(F)⁽¹⁾, . . . Ĥ_(F) ^((U))]; Ĥ_(F) ^((i))=(Ĥ^((i)){circumflex over(F)}^((i)))/√{square root over (2σ_(η,) _(i) ²)} is a combined channelmatrix of a virtual reverse MIMO channel from the ith receiver;Ĥ^((i)=[H) ^((i))]′ is a matrix of the channel state information of thevirtual Reverse MIMO channel from the ith receiver; H^((i)) is a matrixof the channel state information for forward MIMO channel of a pluralityof channels to the ith receiver; {square root over (F)}^((i)) is aspatial coding matrix of the virtual reverse MIMO channel from the ithreceiver; I_(N) _(i) is a unit matrix; N_(i) is a number of signals inthe plurality of groups of at least one signal to be transmitted to theith receiver; and σ_(η,) _(i) ² is a variance of a component of noise atthe ith receiver.
 6. The method of claim 5, further comprising:transmitting a respective set of demodulation weights Ĝ^((i)) to each ofthe receivers.
 7. The method of claim 5, wherein {circumflex over(F)}^((i))= V ^((i))Φ^((i)) where V ^((i)) is a matrix constructed fromcolumns of V^((i)); V^((i)) is a unitary matrix resulting from asingular decomposition of the channel matrix H^((i)) of the forward MIMOchannel to the ith receiver as {tilde over(H)}^((i))=U^((i))Λ^((i))V^((i)) ^(H) , where U^((i)) and V^((i)) areunitary matrices, Λ^((i)) is a non-negative diagonal matrix, squares ofdiagonal elements of Λ^((i)) are equal to eigenvalues of anĤ^((i))Ĥ^((i)′) matrix, columns of U^((i)) are eigenvectors of theĤ^((i))Ĥ^((i)′) matrix, and columns of V^((i)) are also eigenvectors ofthe Ĥ^((i))Ĥ^((i)′) matrix; and Φ^((i)) is a diagonal matrix havingnon-negative diagonal elements that determine channel power allocationand satisfy${{{tr}\left( {{\hat{F}}^{(i)}{{\hat{F}}^{(i)}}^{\prime}} \right)} = {{\sum\limits_{k = 1}^{K_{{ch},i}}\;{\phi_{k,k}^{(i)}}^{2}} = P_{s}}},$ where K_(ch,i) is a number of spatial channels to the ith receiver. 8.The method of claim 7, wherein the diagonal elements of Φ^((i)) areselected according to a criterion selected from a group consisting of: auniform power criterion, φ^((i))k,k²=P_(s)/K_(ch,i); an MMSE (MaximumMean Squared Error) criterion,${{\phi_{k,k}^{(i)}}^{2} = {2{{\sigma_{\eta,i}}^{2}\left\lbrack {\frac{\mu}{\sqrt{\xi_{k,k}^{(i)}}} - \frac{1}{\xi_{k,k}^{(i)}}} \right\rbrack}^{+}}};$an MSER (Minimum Symbol-Error-Rate) criterion,${{\phi_{k,k}^{(i)}}^{2} = {\frac{2{\sigma_{\eta,i}}^{2}}{\xi_{k,k}^{(i)}}\left\lbrack {{\log\left( \frac{\xi_{k,k}^{(i)}}{2{\sigma_{\eta,i}}^{2}} \right)} - \mu} \right\rbrack}^{+}};$and an MCIR (Maximum Capacity and Information Rate) criterion,${\phi_{k,k}^{{(i)}2} = \left( {\mu - \frac{2\sigma_{\eta,i}^{2}}{\xi_{k,k}^{(i)}}} \right)^{+}},$where${(\bullet)^{+} = {{\max\left( {\bullet,0} \right)} = {\frac{1}{2}\left( {{\bullet } + \bullet} \right)}}};$ξ^((i)) _(k,k)=Λ^((i)) _(k,k) ² are eigenvalues of the Ĥ^((i))Ĥ^((i)′)matrix, and Λ^((i)) _(k,k) are diagonal elements of the Λ^((i)) matrix;and μ is a factor selected to define the MMSE, MSER, and MCIR criteria.